Calculus, originally called infinitesimal calculus or “the calculus of infinitesimals”, is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.
It has two major branches, differential calculus and integral calculus; the former concerns instantaneous rates of change, and the slopes of curves, while integral calculus concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus, and they make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit.
“Calculus – Wikipedia”. 2021. en.wikipedia.org. https://en.wikipedia.org/wiki/Calculus.
Learning Calculus
I would recommend learning calculus in the following order as they build a foundation in understanding calculus as you progress. Information on each topic can be found in the online book Boundless Calculus. You may also want to checkout other Textbook Solutions & Explanations web pages to assist you in your studies of calculus.
Building Blocks of Calculus
Precalculus Review
Functions and Models
Limits
Derivatives and Integrals
Derivatives
Applications of Differentiation
Integrals
Applications of Integration
Inverse Functions and Advanced Integration
Inverse Functions: Exponential, Logarithmic, and Trigonometric Functions
Techniques of Integration
Further Applications of Integration
Differential Equations, Parametric Equations, and Sequences and Series
Differential Equations
Parametric Equations and Polar Coordinates
Infinite Sequences and Series
Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus
Vectors and the Geometry of Space
Vector Functions
Partial Derivatives
Multiple Integrals
Vector Calculus
Second-Order Linear Equations
Newton v/s Leibniz, the Great Calculus Controversy
Both Newton and Leibniz had arrived at pretty much the same mathematical grounds, the actual controversy was: who actually invented it? Newton was a Lucasian professor of mathematics at Cambridge University and his supporters would often claim and accuse Leibniz of stealing Newton’s works claiming that he observed Newton’s papers. This started a heated argument between the two geniuses and their supporters. Leibniz was absolutely disgraced by Newton’s accusations of plagiarism. While Leibniz reportedly admitted that he was shown Newton’s work, the research of historians shows that his approach to developing the mathematical foundations of calculus was different than that of Newton’s and Leibniz has explained that in his papers. Despite that fact, Leibniz had to put up with Newton’s vicious attacks.
Labh, Sunny. “Newton v/s Leibniz, the Great Calculus Controversy”. 2022. Medium. https://piggsboson.medium.com/newton-v-s-leibnitz-the-great-calculus-controversy-efb2bc97c23e.
Calculus Challenges
To really challenge yourself and get a better understanding of differential and integral calculus, use the following Instagram accounts. The equations they present will test your knowledge and assist you to learn calculus.
“calculus4you”. 2022. Instagram. https://www.instagram.com/calculus4you/.
“highermaths”. 2022. Instagram. https://www.instagram.com/highermaths/.
Recommended Reading
“Articles From Calculus”. 2022. dummies.com. https://www.dummies.com/category/articles/calculus-33723.
⭐ Brahambhatt, Rupendra, Derya Ozdemir, Loukia Papadopoulos, Maia Mulko, and Matthew Williams. 2022. “What Is Calculus And Does It Serve Any Purpose In The Real World?” interestingengineering.com. https://interestingengineering.com/calculus.
Can you think of your life where nothing changed? Those moments are rare, as everything in our life is subject to transformation. The amount of food we eat every day, the number of steps we take, and even the time the Sun rises. Calculus is a branch of mathematics that allows us to study continuous change. Most people see calculus only as a bunch of equations that involve a lot of calculations, but it is actually the set of principles that we apply in our lives every day.
The application of calculus can be seen in physics, medicine, engineering, and economics. It played a key role in the development of navigation in the 17th and 18th centuries, and today it has a vital role in space travel, and in the development of various other advanced technologies. The fields and disciplines that use calculus include thermodynamics, electricity, acoustics, geography, computer vision, economics, robotics, demographics, ship design, and engineering.
“Calculus, Better Explained Ebook + Video Course”. 2021. betterexplained.com. https://betterexplained.com/calculus/.
This is the calculus primer I wish I had in school. Learn the essential concepts using concrete analogies and vivid diagrams, not mechanical definitions. Calculus isn’t a set of rules, it’s a specific, practical viewpoint we can apply to everyday thinking.
“Calculus – Formula, Definition, Examples | What Is Calculus?”. 2021. Cuemath. https://www.cuemath.com/calculus/.
⭐ “How Infinite Series Reveal The Unity Of Mathematics | Quanta Magazine”. 2022. Quanta Magazine. https://www.quantamagazine.org/how-infinite-series-reveal-the-unity-of-mathematics-20220124/.
Infinite sums are among the most underrated yet powerful concepts in mathematics, capable of linking concepts across math’s vast web.
“Introduction To Calculus”. 2021. mathsisfun.com. https://www.mathsisfun.com/calculus/introduction.html.
“Introduction To Calculus”. 2021. Medium. https://medium.com/math-simplified/introduction-to-calculus-356fe498f2b3.
Ming, Albert. “Why You Should Learn Calculus”. 2021. Medium. https://medium.com/age-of-awareness/why-you-should-learn-calculus-eb4b325136ce.
Since its creation, calculus has undoubtedly changed the world of mathematics for the better. It is a brilliant and new way of thinking that explores the principles of continuous change. You may have heard of some of calculus’s many topics, including differentiation, integration, and so on. But have you ever considered how impactful these concepts really are? Many people end up just cramming techniques into their heads, whether it be the power rule or integration by parts, without actually understanding how useful they can be. However, I am here to tell you that calculus has impacts in all sorts of areas, from economics all the way to industrial design. These impacts have heavily aided researchers in their respective areas of work.
Ryan, Mark. Calculus For Dummies. Hoboken, NJ. John Wiley & Sons, Inc., 2014.
⭐ Strogatz, Steven. Infinite Powers: How Calculus Reveals the Secrets of the Universe. New York: Haughton Mifflin Harcourt, 2019.
“The Six Pillars Of Calculus”. 2022. web.ma.utexas.edu. https://web.ma.utexas.edu/users/m408n/m408c/CurrentWeb/LM0-0-1.php.
Calculus is generally viewed as a difficult subject, with hundreds of formulas to memorize and many applications to the real world. However, almost all of calculus boils down to six basic ideas, together with one precalculus idea. If you really understand these ideas, the formulas will come easily. If you don’t understand these ideas, the formulas won’t make any sense. We’ll develop some of the ideas slowly.
“The Thrilling Story Of Calculus”. 2022. Medium. https://thewalkingtemple.medium.com/the-thrilling-story-of-calculus-5853ad7a615c.
“Why Is Calculus Hard?” 2021. Medium. https://medium.com/@chrmn.arc/why-is-calculus-hard-4c964a64c863.
Will, Karen. “What Are The Practical Uses Of Calculus, Other Than Calculating The Area Under Curves?”. 2021. Quora. https://www.quora.com/What-are-the-practical-uses-of-calculus-other-than-calculating-the-area-under-curves.
Architecture
Unbeknownst to many math students, calculus is one of the backbones of architecture. Particularly in bridge-building, calculus must be used to make complex constructions in order to support the weight across vast spaces.
Business
In the world of business, calculus is primarily used for optimization. Whether this means building profits or undercutting costs, calculus is used to analyze the rate of change.
Engineering
Calculus is also a tool used in the field of engineering. Aerospace engineers love the Navier-Stokes equations, an invaluable description of what happens in the physical realm when aircraft takes flight.
Economics
Because of its connection to variables, derivatives, and rates of change, calculus is used widely in economics to forecast supply and demand, as well as how to reap maximum profits.
Search Engines
Calculus in search engines? Yes, used as algorithms. Algorithms are the basic calculations that fit large portions of data into an equation. When an algorithm comes to play in an online search, they absorb and utilize information such as the user’s location, web history, and the usefulness of web search results.
Weather
When meteorologists predict the weather, the technology of computer modeling comes into play. Weather models use programs on computers that contain algorithm types to help assign all the possible weather outcomes in a specific region.
Public Health
Epidemiology, or the study of how infectious disease is spread, depends on the use of calculus. The three main factors one must consider in epidemiology are 1) people prone to the disease, 2) those who have recovered from the disease, and 3) those who have the disease. Calculus, making use of these three variables, can tell us the mass effect of a disease – how far it is spreading, how fast it is spreading, and the best way to treat it.
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Videos
Don’t Memorise. “Calculus – Basics”. 2021. youtube.com. https://www.youtube.com/playlist?list=PLmdFyQYShrjd4Qn42rcBeFvF6Qs-b6e-L.
Calculus can be referred to as the mathematics of change. In this learning playlist, you are going to understand the basic concepts of calculus, so you can develop the skill of predicting the change.
These videos will help you understand Limits, Differentiation and Integration- the important ideas around which calculus is built. In calculus, we mostly study things which change according to a pattern. These patterns of change are mathematically represented by the concept of functions. Since the concept of functions is the main object of study in calculus, this playlist will also focus on helping you gain a deep understanding of it. Finally, this playlist will teach you how to use the ideas of limits, differentiation and integration to analyse functions.
If you want to grow your knowledge of mathematics, watch this learning playlist!
⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.
Mathematical Mysteries 